Sum of the series `a^n+a^(n-1)b+^(n-2)b^2+………..+ab^n` can be obtained by taking outt `a^n or b^n` comon and using the forumula of sum of `(n+1)` terms of G.P. N the basis of above information answer the following question: Coefficient of `x^50 in (1+x)^1000+x(1+x)^999+........+x^999(1+x)+x^1000 ` is (A) `^1000C_50` (B) `^1002C_50` (C) `^1001C_50` (D) `^1001C_49`

Sum of the series a^n+a^(n-1)b+^(n-2)b^2+………..+ab^n can be obtained by taking outt a^n or b^n comon and using the forumula of sum of (n+1) terms of G.P. answer the following question:Um of coeficients of x^50 (1+x)^1000 +x(1+x)^999+....

Find the coefficients of x^(50) in the expression (1+x)^(1000)+2x(1+x)^(999)+3x^2(1+x)^(998)+....+1001 x^(1000) .

Let a,b,c,d be the four consecutive coefficients of the binomial expansion (1+x)^n On the basis of above information answer the following question: a/(a+b), b/(b+c), c/(c+d) are in (A) A.P. (B) G.P. (C) H.P. (D) none of these

If the coefficient of x^100 is in 1+(1+x)+(1+x)^2+(1+x)^3+………+(1+x)^n , (n >= 100) is ^201C_101 then n (A) 100 (B) 200 (C) 101 (D) none of these

If n is a positive integer then ^nC_r+^nC_(r+1)=^(n+1)^C_(r+1) Also coefficient of x^r in the expansion of (1+x)^n=^nC_r. In an identity in x, coefficient of similar powers of x on the two sides re equal. On the basis of above information answer the following question: If n is a positive integer then ^nC_n+^(n+1)C_n+^(n+2)C_n+.....+^(n+k)C_n= (A) ^(n+k+1)C_(n+2) (B) ^(n+k+1)C_(n+1) (C) ^(n+k+1)C_k (D) ^(n+k+1)C_(n-2)

The coefficient of x^50 in the polynomial (x + ^50C_0)(x +3.^5C_1) (x +5.^5C_2).....(x + (2n + 1) ^5C_50) , is

If n is a positive integer such that (1+x)^n=^nC_0+^nC_1+^nC_2x^2+…….+^nC_nx^n , for epsilonR . Also .^nC_r=C_r On the basis of the above information answer the following questions The value of the expression a-(a-1)C_1+(a-2)C_2-(a-3)C_3+.......+(1)^n(a-n)C_n= (A) 0 (B) a^n.(-1)^n.^(2n)C_n (C) [2a-n(n+1)].^(2n)C_n (D) none of these

If n is a positive integer such that (1+x)^n=^nC_0+^nC_1+^nC_2x^2+…….+^nC_nx^n , for epsilonR . Also .^nC_r=C_r On the basis of the above information answer the following questions The value of the series sum_(r=1)^n r^2.C_r= (A) 1 (B) (-1)^(n/2).(n!)/(n/(2!))^2 (C) (n-1).^(2n)C_n+2(2n) (D) n(n+1)2^(n-2)

If n is a positive integer then ^nC_r+^nC_(r+1)=^(n+1)^C_(r+1) Also coefficient of x^r in the expansion of (1+x)^n=^nC_r. answer the following question: If n is a positive integer then ^nC_n+^(n+1)C_n+^(n+2)C_n+.....+^(n+k)C_n= (A) ^(n+k+1)C_(n+2) (B) ^(n+k+1)C_(n+1) (C) ^(n+k+1)C_k (D) ^(n+k+1)C_(n-2)

In the expansion of (1+x)^50 the sum of the coefficients of odd power of x is (A) 0 (B) 2^50 (C) 2^49 (D) 2^51